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Those Fascinating Numbers
Page236(256 of 451)

236 Jean-Marie De Koninck 103 511 • the 10 000th prime power, in fact here a prime (see the number 419). 104 729 • the 10 000th prime number (see the number 541). 105 212 • the number of Carmichael numbers 1015 (see the number 646). 106 755 • the smallest six digit Sastry number (see the number 6 099). 106 853 • the smallest prime number q such that ∑ p≤q p is a multiple of 10 000: here ∑ p≤106 853 p = 515 530 000 (see the number 35 677). 109 297 • the eighth number k such that 11 . . . 1 k is prime (discovered by Bourdelais and Dubner in 2007); see the number 19. 109 306 • the third number which is not a cube, but which can be written as the sum of the cubes of some of its prime factors: here 109 306 = 2·31·41·43 = 23+313+433 (see the number 378, as well as 870); besides the numbers which can be written as the sum of the cubes of their prime factors (see the number 378 for a list of eight such numbers) and besides the number 109 306, the following numbers also satisfy this property172: 23 391 460 = 22 · 5 · 23 · 211 · 241 = 23 + 2113 + 2413, 173 871 316 = 22 · 223 · 421 · 463 = 23 + 4213 + 4633, 450 843 455 098 = 2 · 6007 · 6089 · 6163 = 23 + 60073 + 61633. 172It is easy to prove that if Hypothesis H is true, then there exist infinitely many such numbers. Indeed, this follows from the fact that under this hypothesis, there exist infinitely many even numbers k such that the two numbers r = k2 − 9k + 21 and p = k2 − 7k + 13 are prime, in which case by setting n = 2rqp, where q = k2 − 8k + 21, it is easy to prove that n = 23 + r3 + p3. This result as well as other results on this topic can be found in De Koninck & Luca [54].